Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2018  FebMar  (P19709/12)  Q#8
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Question
A curve has equation .
i.Find the xcoordinates of the stationary points.
ii.Find .
iii.Find, showing all necessary working, the nature of each stationary point.
Solution
i.
We are required to find the coordinates of the stationary point of the curve.
Coordinates of stationary point on the curve can be found from the derivative of equation of the curve by equating it with ZERO. This results in value of xcoordinate of the stationary point
Therefore, first we find the derivative of equation of the curve.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve
We are given equation of the curve as;
Therefore;
Rule for differentiation of
Rule for differentiation of
Now, to find the coordinates of stationary point of the curve;
Let
Now we have two options.




Since 





ii.
Second derivative is the derivative of the derivative. If we have derivative of the curve
We have found in (i) that;
Therefore;
Rule for differentiation of
Rule for differentiation of
Rule for differentiation of
iii.
Next we are required to find the nature of each stationary point.
Once we have the xcoordinate of the stationary point
We substitute
If
If
From (i) we have xcoordinates of s a stationary points on the curve as 16 & 4.
Therefore, substituting
Since
Next, substituting
Since
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